you could filter the 20 numbers to look for 3 odd 3 even or 4 odd 2 even to save time
add
$oddn = ($df.Split(",") | % {if($_ % 2 -eq 1 ) {"$_" | sort} }) -join (",") # odd
$evenn = ($df.Split(",") | % {if($_ % 2 -eq 0 ) {"$_"| sort} }) -join (",") #even
$even1 = $evenn.Split(",")
$odd1 = $oddn.Split(",")
$countod = @($odd1).Count
$countev = @($even1).Count
and do a if statement and a little more code
the the total 12320 combinations of 6 numbers from 20 numbers 3 odd 3 even
1,2,3,4,5,6,,,odd,3,,,1,3,5,,,even,3,,,2,4,6
1,2,3,4,5,8,,,odd,3,,,1,3,5,,,even,3,,,2,4,8
1,2,3,4,5,10,,,odd,3,,,1,3,5,,,even,3,,,2,4,10
1,2,3,4,5,12,,,odd,3,,,1,3,5,,,even,3,,,2,4,12
1,2,3,4,5,22,,,odd,3,,,1,3,5,,,even,3,,,2,4,22
1,2,3,4,5,44,,,odd,3,,,1,3,5,,,even,3,,,2,4,44
1,2,3,4,6,7,,,odd,3,,,1,3,7,,,even,3,,,2,4,6
1,2,3,4,6,9,,,odd,3,,,1,3,9,,,even,3,,,2,4,6
1,2,3,4,6,11,,,odd,3,,,1,3,11,,,even,3,,,2,4,6
1,2,3,4,6,13,,,odd,3,,,1,3,13,,,even,3,,,2,4,6
1,2,3,4,6,15,,,odd,3,,,1,3,15,,,even,3,,,2,4,6
1,2,3,4,6,21,,,odd,3,,,1,3,21,,,even,3,,,2,4,6
1,2,3,4,6,23,,,odd,3,,,1,3,23,,,even,3,,,2,4,6
1,2,3,4,6,27,,,odd,3,,,1,3,27,,,even,3,,,2,4,6
1,2,3,4,6,45,,,odd,3,,,1,3,45,,,even,3,,,2,4,6
1,2,3,4,7,8,,,odd,3,,,1,3,7,,,even,3,,,2,4,8
1,2,3,4,7,10,,,odd,3,,,1,3,7,,,even,3,,,2,4,10
1,2,3,4,7,12,,,odd,3,,,1,3,7,,,even,3,,,2,4,12
1,2,3,4,7,22,,,odd,3,,,1,3,7,,,even,3,,,2,4,22
1,2,3,4,7,44,,,odd,3,,,1,3,7,,,even,3,,,2,4,44
1,2,3,4,8,9,,,odd,3,,,1,3,9,,,even,3,,,2,4,8
1,2,3,4,8,11,,,odd,3,,,1,3,11,,,even,3,,,2,4,8
1,2,3,4,8,13,,,odd,3,,,1,3,13,,,even,3,,,2,4,8
1,2,3,4,8,15,,,odd,3,,,1,3,15,,,even,3,,,2,4,8
1,2,3,4,8,21,,,odd,3,,,1,3,21,,,even,3,,,2,4,8
1,2,3,4,8,23,,,odd,3,,,1,3,23,,,even,3,,,2,4,8
1,2,3,4,8,27,,,odd,3,,,1,3,27,,,even,3,,,2,4,8
1,2,3,4,8,45,,,odd,3,,,1,3,45,,,even,3,,,2,4,8
1,2,3,4,9,10,,,odd,3,,,1,3,9,,,even,3,,,2,4,10
1,2,3,4,9,12,,,odd,3,,,1,3,9,,,even,3,,,2,4,12
1,2,3,4,9,22,,,odd,3,,,1,3,9,,,even,3,,,2,4,22
1,2,3,4,9,44,,,odd,3,,,1,3,9,,,even,3,,,2,4,44
1,2,3,4,10,11,,,odd,3,,,1,3,11,,,even,3,,,2,4,10
1,2,3,4,10,13,,,odd,3,,,1,3,13,,,even,3,,,2,4,10
1,2,3,4,10,15,,,odd,3,,,1,3,15,,,even,3,,,2,4,10
1,2,3,4,10,21,,,odd,3,,,1,3,21,,,even,3,,,2,4,10
1,2,3,4,10,23,,,odd,3,,,1,3,23,,,even,3,,,2,4,10
1,2,3,4,10,27,,,odd,3,,,1,3,27,,,even,3,,,2,4,10
1,2,3,4,10,45,,,odd,3,,,1,3,45,,,even,3,,,2,4,10
1,2,3,4,11,12,,,odd,3,,,1,3,11,,,even,3,,,2,4,12
1,2,3,4,11,22,,,odd,3,,,1,3,11,,,even,3,,,2,4,22
1,2,3,4,11,44,,,odd,3,,,1,3,11,,,even,3,,,2,4,44
1,2,3,4,12,13,,,odd,3,,,1,3,13,,,even,3,,,2,4,12
1,2,3,4,12,15,,,odd,3,,,1,3,15,,,even,3,,,2,4,12
1,2,3,4,12,21,,,odd,3,,,1,3,21,,,even,3,,,2,4,12
1,2,3,4,12,23,,,odd,3,,,1,3,23,,,even,3,,,2,4,12
1,2,3,4,12,27,,,odd,3,,,1,3,27,,,even,3,,,2,4,12
1,2,3,4,12,45,,,odd,3,,,1,3,45,,,even,3,,,2,4,12
1,2,3,4,13,22,,,odd,3,,,1,3,13,,,even,3,,,2,4,22
1,2,3,4,13,44,,,odd,3,,,1,3,13,,,even,3,,,2,4,44
1,2,3,4,15,22,,,odd,3,,,1,3,15,,,even,3,,,2,4,22
1,2,3,4,15,44,,,odd,3,,,1,3,15,,,even,3,,,2,4,44
1,2,3,4,21,22,,,odd,3,,,1,3,21,,,even,3,,,2,4,22
1,2,3,4,21,44,,,odd,3,,,1,3,21,,,even,3,,,2,4,44
1,2,3,4,22,23,,,odd,3,,,1,3,23,,,even,3,,,2,4,22
1,2,3,4,22,27,,,odd,3,,,1,3,27,,,even,3,,,2,4,22
1,2,3,4,22,45,,,odd,3,,,1,3,45,,,even,3,,,2,4,22
1,2,3,4,23,44,,,odd,3,,,1,3,23,,,even,3,,,2,4,44
1,2,3,4,27,44,,,odd,3,,,1,3,27,,,even,3,,,2,4,44
1,2,3,4,44,45,,,odd,3,,,1,3,45,,,even,3,,,2,4,44
1,2,3,5,6,8,,,odd,3,,,1,3,5,,,even,3,,,2,6,8